Observational cosmology: TheFriedmanequations1 -...

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Observational Observational cosmology: cosmology: The Friedman equations 1 The Friedman equations 1 Filipe B. Abdalla Filipe B. Abdalla Kathleen Lonsdale Building G.22 Kathleen Lonsdale Building G.22 http://zuserver2.star.ucl.ac.uk/~hiranya/PHAS3136/PHAS3136 http://zuserver2.star.ucl.ac.uk/~hiranya/PHAS3136/PHAS3136

Transcript of Observational cosmology: TheFriedmanequations1 -...

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ObservationalObservationalcosmology:cosmology:

The Friedman equations 1The Friedman equations 1

Filipe B. AbdallaFilipe B. AbdallaKathleen Lonsdale Building G.22Kathleen Lonsdale Building G.22

http://zuserver2.star.ucl.ac.uk/~hiranya/PHAS3136/PHAS3136http://zuserver2.star.ucl.ac.uk/~hiranya/PHAS3136/PHAS3136

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A brain teaser: The anthropic principle!A brain teaser: The anthropic principle!•• Last lecture I said “Is cosmology a science given that weLast lecture I said “Is cosmology a science given that we

only have one Universe?”only have one Universe?”•• Weak anthropic principle:Weak anthropic principle: "The observed values of all"The observed values of all

physical andphysical and cosmologicalcosmological quantities are not equallyquantities are not equallyprobable but they take on values restricted by theprobable but they take on values restricted by therequirement that there exist sites whererequirement that there exist sites where carboncarbon--based lifebased lifecancan evolveevolve and by the requirements that the Universe beand by the requirements that the Universe beold enough for it to have already done so "old enough for it to have already done so "old enough for it to have already done so.old enough for it to have already done so.

•• Strong anthropic principleStrong anthropic principle: "The Universe must have: "The Universe must havethose properties which allow life to develop within it atthose properties which allow life to develop within it atsome stage in its history.“some stage in its history.“

•• Does it make sense to assume we exist and inferDoes it make sense to assume we exist and inferfundamental values for components of the Universe?fundamental values for components of the Universe?–– Can we say anything about Lambda given that we exist?Can we say anything about Lambda given that we exist?–– Can we say anything about the matter density given thatCan we say anything about the matter density given that

we exist?we exist?•• If one could make a second Universe how do we knowIf one could make a second Universe how do we know

there would be life?there would be life?

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OUTLINE:OUTLINE:Friedman equation derivationFriedman equation derivationAfter the lecture, you should be able to:After the lecture, you should be able to:•• Define/derive the terms: “metric”, “scale factor”Define/derive the terms: “metric”, “scale factor”

and “coand “co--moving coordinates”moving coordinates”•• Derive the relation between the scale factor andDerive the relation between the scale factor and

redshiftredshift•• Derive the Friedman equation using NewtonianDerive the Friedman equation using Newtonian

argumentsarguments•• Describe and discuss the possible geometries ofDescribe and discuss the possible geometries of

the Universethe Universe•• [Non[Non--examinable: Discuss possible topologies ofexaminable: Discuss possible topologies of

the Universe]the Universe]

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Our world line in special relativity!Our world line in special relativity!

•• Is a one dimensional line orIs a one dimensional line orcurve that represents thecurve that represents thecoordinates of a given place incoordinates of a given place inspacespace--time.time.

•• As an objects moves the worldAs an objects moves the worldline moves sideways. As timeline moves sideways. As timeyypasses a static object movespasses a static object movesalong the z axis.along the z axis.

•• Einstein said v<c. so worldEinstein said v<c. so worldlines don’t bend more than 45lines don’t bend more than 45degrees or x/t > c.degrees or x/t > c.

•• The photon world line definesThe photon world line definesboundaries of the knowableboundaries of the knowableUniverse from the unknowableUniverse from the unknowableUniverseUniverse

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The metric.The metric.•• In relativity space and time are mixed up so we have toIn relativity space and time are mixed up so we have to

define a distance which defines how separate 2 events aredefine a distance which defines how separate 2 events aredistant from one another in spacedistant from one another in space--time.time.

•• Infinitesimally separated events in space and time have aInfinitesimally separated events in space and time have adistance equal to:distance equal to:

•• This is a spaceThis is a space--time metric. It determines who one counttime metric. It determines who one countdistance between 2 points.distance between 2 points.

•• If we measure distances from the origin there is no harmIf we measure distances from the origin there is no harmin choosing a spherical polar coordinate system:in choosing a spherical polar coordinate system:

•• So the proper time difference between two events isSo the proper time difference between two events is

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The metric.The metric.•• For light:For light:•• IfIf

the interval is a space like eventthe interval is a space like event•• IfIfIfIf

the interval is a time like intervalthe interval is a time like interval•• If spaceIf space--time is curved then the metrictime is curved then the metric

defines the straightest possible world line.defines the straightest possible world line.I.e. the geodesic. It is defined by:I.e. the geodesic. It is defined by:

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Euclidean geometryEuclidean geometry•• = “flat” geometry= “flat” geometry

•• parallel straight lines never meetparallel straight lines never meet•• triangle angles add up to 180triangle angles add up to 180°°•• circumference of circle = 2circumference of circle = 2ππ rr•• circumference of circle = 2circumference of circle = 2ππ rr•• NB. General definition of a straight line:NB. General definition of a straight line:

–– shortest distance between two pointsshortest distance between two points–– applies for nonapplies for non--flat geometriesflat geometries

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The surface of a objects…The surface of a objects…•• Are lines of longitude straight?Are lines of longitude straight?

–– Yes, they are great circlesYes, they are great circles

•• Are lines of latitude straight?Are lines of latitude straight?–– No, they are not great circlesNo, they are not great circles

•• Is the surface of a sphere flat?Is the surface of a sphere flat?–– NoNo

•• Is the surface of a cylinder flat?Is the surface of a cylinder flat?–– YesYes

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The metric of a sphereThe metric of a sphere

•• The equation for a sphere:The equation for a sphere:•• So if we calculateSo if we calculate

differentialsdifferentials•• This leads toThis leads to•• If we go to a moreIf we go to a more

convenientconvenientparameterisationparameterisation

•• This means the spaceThis means the space--likelikeelement is:element is:

•• Notice the factorNotice the factorunderneath the radial termsunderneath the radial terms

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The metric of a sphereThe metric of a sphereembedded in 4Dembedded in 4D•• The equation for a 3 sphereThe equation for a 3 sphere

embedded in 4D:embedded in 4D:•• So if we calculateSo if we calculate

differentialsdifferentials•• This leads toThis leads to•• This leads toThis leads to•• If we go to a moreIf we go to a more

convenientconvenientparameterisationparameterisation

•• This means the spaceThis means the space--likelikeelement is:element is:

•• Notice the factorNotice the factorunderneath the radial termsunderneath the radial terms… again!!!!!… again!!!!!we define k = 1/a^2we define k = 1/a^2

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Summary of GeometriesSummary of Geometries

CurvatureCurvature GeometryGeometry Angles ofAngles oftriangletriangle

CircumferenceCircumferenceof circleof circle

Type ofType ofUniverseUniverse

k > 0k > 0 sphericalspherical >180>180°° c < 2c < 2 ππ rr ClosedClosed

• Copy of Liddle Table 4.2:

k = 0k = 0 flatflat 180180°° c = 2c = 2 ππ rr FlatFlat

k < 0k < 0 hyperbolichyperbolic <180<180°° c > 2c > 2 ππ rr OpenOpen

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The meaning of k inThe meaning of k in

•• k = 1 /Rk = 1 /R22 (from GR), where R = radius of curvature(from GR), where R = radius of curvature•• FlatFlat

–– R = infinityR = infinity–– k = 0k = 0 /c

sflevel1/cffig1.jpg

•• SphericalSpherical–– R < infinityR < infinity–– k > 0k > 0

•• HyperbolicHyperbolic–– R imaginaryR imaginary–– k < 0k < 0

http://www.dur.ac.uk/Physics/students/

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The FRW metric:The FRW metric:•• We want to be general… i.e. for our Universe weWe want to be general… i.e. for our Universe we

want to write a metric which is isotropic.want to write a metric which is isotropic.

•• For a homogeneous and isotropic Universe weFor a homogeneous and isotropic Universe wecan prove that (from GR later…)can prove that (from GR later…)

•• k can be associated with the Gaussian curvaturek can be associated with the Gaussian curvatureof the Universe…of the Universe…

•• Fro convenience we canFro convenience we canchange variables to:change variables to:

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The derivation of the redsfhitThe derivation of the redsfhit•• For a photon, using the FRW metric we have:For a photon, using the FRW metric we have:

•• How if we take a photon arrival and emissionHow if we take a photon arrival and emissiontime:time:

•• The second equality is because we can take theThe second equality is because we can take thesecond crest given that the photon is a wave.second crest given that the photon is a wave.

•• So equating the two we haveSo equating the two we have

•• If we assume a(t) is unchanging during theseIf we assume a(t) is unchanging during thesesmall intervals we can take a out of the integral.small intervals we can take a out of the integral.

•• Hence:Hence:

•• So if we take the time intervals to be the periodsSo if we take the time intervals to be the periodsof the photon we have now:of the photon we have now:

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Principle of Equivalence:Principle of Equivalence:

•• Constant ratio of the inertial massConstant ratio of the inertial massand the gravitational massand the gravitational mass

•• Means that “All local freely falling,Means that “All local freely falling,nonnon--rotating labs are fullyrotating labs are fully

i l t f th fi l t f th fequivalent for the purposes ofequivalent for the purposes ofphysical experiments”physical experiments”

•• i.e. the strong equivalent principle ti.e. the strong equivalent principle tis always possible to choose a localis always possible to choose a localcoco--ordinate system such that all theordinate system such that all thelaws of physics have the same form.laws of physics have the same form.

•• Acceleration = gravitation =Acceleration = gravitation =curvaturecurvature

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Derive the Friedmann equationDerive the Friedmann equation•• Describes expansion rate of UniverseDescribes expansion rate of Universe•• a = scale factora = scale factor•• da/dt = differential of a wrt timeda/dt = differential of a wrt time•• ρρ = matter density= matter density•• ρρ = matter density= matter density•• k = a constantk = a constant

•• Full derivation uses GR.Full derivation uses GR.•• Use Newtonian derivation hereUse Newtonian derivation here

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Friedman eq. derivation:Friedman eq. derivation:•• Lets follow Newton and write the force on aLets follow Newton and write the force on a

mass m.mass m.

•• The particle’s gravitational potential can beThe particle’s gravitational potential can bewritten as:written as:

•• The kinetic energy of a particle can beThe kinetic energy of a particle can bewritten as:written as:

•• Energy conservations gives us:Energy conservations gives us:

•• The relation between the position and theThe relation between the position and thecoco--moving position is:moving position is:

•• So the total internal energy of the system is:So the total internal energy of the system is:•• Making the substitution:Making the substitution:

•• We finally have: (we will prove that this isWe finally have: (we will prove that this isthe Hubble constant later)the Hubble constant later)

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General Relativity in two slides:General Relativity in two slides:Christophel, Ricci and Riemann (non examinable)Christophel, Ricci and Riemann (non examinable)

• All of Special Relativity applies.• Worldlines are straight.• In a non-inertial frame, there are accelerations:

where the Christophelsymbol is:

and the Riemann tensor and Ricci tensor and scalars aredefined as:

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General Relativity in two slides:General Relativity in two slides:Einstein……….. (non examinable)Einstein……….. (non examinable)

• Einstein told us that

and also that

• For perfect fluids

where U is the 4 momentum, and these Einsteinequations reduce to 2Friedman equationscosmologists use…

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Implications of Friedmann eqnImplications of Friedmann eqn

•• Allows us to solve for a(t)Allows us to solve for a(t)

•• If we knowIf we know ρρ(t) and k(t) and k

•• ρρ(t) depends on contents of Universe(t) depends on contents of Universe–– see next lecture on fluid equationsee next lecture on fluid equation

•• What is the meaning of k?What is the meaning of k?–– From GR derivationFrom GR derivation–– geometry of the Universegeometry of the Universe What is the meaning of k?What is the meaning of k?

•• In GR we can sa we want a maximally symetric metric onlyIn GR we can sa we want a maximally symetric metric onlydependent on the curvature sodependent on the curvature so

•• Implies that the metric should be the way it was derived beforeImplies that the metric should be the way it was derived before

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Topologies:Topologies:•• So far we look at a patch of theSo far we look at a patch of the

Universe but is the Universe infiniteUniverse but is the Universe infiniteor a tiling of bits?or a tiling of bits?

•• One example:One example:–– What kind of Universe is this?What kind of Universe is this?–– A Thorus:A Thorus:

•• Other examples: a bit weird butOther examples: a bit weird butpppossible! Called a torus of genus 2possible! Called a torus of genus 2

•• Simpler but still unusual: what isSimpler but still unusual: what isthis?this?

•• One consequence: we can seeOne consequence: we can seeourselves in the past if we look farourselves in the past if we look faraway enough!!!!!!! We can test thisaway enough!!!!!!! We can test thisseeing circles in the CMBseeing circles in the CMB

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OUTLINEOUTLINEAcceleration equation derivationAcceleration equation derivationAfter the lectures, you should be able to:After the lectures, you should be able to:•• Find the change in density as a function of scale factorFind the change in density as a function of scale factorρρ(a) for matter dominated universe, just by considering(a) for matter dominated universe, just by consideringconservation of matterconservation of matter

•• Derive the fluid equationDerive the fluid equation•• Derive the change in density as a function of scale factorDerive the change in density as a function of scale factorρρ(a) for a single fluid universe with given equation of(a) for a single fluid universe with given equation ofstate, from the fluid equationstate, from the fluid equation

•• Derive the acceleration equation from the FriedmannDerive the acceleration equation from the Friedmannequation and the fluid equationequation and the fluid equation

•• Discuss the cosmological constant and dark energyDiscuss the cosmological constant and dark energy

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The fluid equationThe fluid equation•• The change of volume as aThe change of volume as a

function of time can be written as:function of time can be written as:

•• But if we write the energy as:But if we write the energy as:

•• So the change in energy as aSo the change in energy as afunction of time is:function of time is:

•• Here we start with the second lawHere we start with the second lawof thermodynamics:of thermodynamics:

•• So we have if we assume theSo we have if we assume theexpansion is reversible, i.e.expansion is reversible, i.e.isentropic:isentropic:

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The acceleration equationThe acceleration equation

•• We take the FriedmanWe take the Friedmanequation in the followingequation in the followingform and differentiate it:form and differentiate it:N b tit t thN b tit t th•• Now we substitute theNow we substitute thevalue of the differentialvalue of the differentialof the density from theof the density from thefluid equation back intofluid equation back intothis equation to get:this equation to get:

•• Simplifying we get:Simplifying we get:

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Derivation for the density evolutionDerivation for the density evolutionfor a component with w.for a component with w.

•• We can reWe can re--write the accelerationwrite the accelerationequation and the fluid equationequation and the fluid equationwith the natural parameter w thewith the natural parameter w theratio of the pressure to the energyratio of the pressure to the energydensitydensitydensity.density.

•• ReRe--writing the fluid equation inwriting the fluid equation insuch way:such way:

•• We can describe how differentWe can describe how differentcomponents with different w evolvecomponents with different w evolveas a function of the scale factor.as a function of the scale factor.

•• So for example we have provenSo for example we have proventhat if a component such as matterthat if a component such as matterhas no pressure, then the densityhas no pressure, then the densityvaries as the cubed power of avaries as the cubed power of a

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Different component evolutions:Different component evolutions:

•• Matter: w = 0 varies as a cubedMatter: w = 0 varies as a cubed•• Radiation w = 1/3 varies as a to the 4Radiation w = 1/3 varies as a to the 4•• A cosmological constant is has rhoA cosmological constant is has rho•• A cosmological constant is has rhoA cosmological constant is has rho

constant so w must beconstant so w must be --1.1.•• Any particle which goes from relativistic toAny particle which goes from relativistic to

nonnon--relativistic has w from 1/3 to 0.relativistic has w from 1/3 to 0.

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Matter/radiation dominationMatter/radiation domination•• In a matter dominatedIn a matter dominated

universe, given that theuniverse, given that thedensity goes as the cube ofdensity goes as the cube ofthe scale factor:the scale factor:

•• Try a solutionTry a solution•• For matter domination weFor matter domination we

have: q=2/3.have: q=2/3.•• So the Hubble parameterSo the Hubble parameter

can be written as :can be written as :•• Implications for the age ofImplications for the age of

the Universe! This value isthe Universe! This value isless than the age of someless than the age of somesystems. Which?systems. Which?

•• For radiation dominationFor radiation dominationq=1/2q=1/2

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The cosmological constantThe cosmological constant•• Einstein spotted a constant of integrationEinstein spotted a constant of integration

•• Appears in our equations:Appears in our equations:

•• Out of fashion (assumed zero) until ~ 10 years agoOut of fashion (assumed zero) until ~ 10 years ago

•• Looks just like a fluid with w=Looks just like a fluid with w=--11

–– has negative pressure!has negative pressure!

•• The vacuum energy from particle physics could produce this effectThe vacuum energy from particle physics could produce this effect

–– current calculations givecurrent calculations give ΛΛ a factor 10a factor 10120120 too high!too high!

–– Compare the energy in dark energy to GUT scale energies…Compare the energy in dark energy to GUT scale energies…

•• ImpliesImplies ρρ(a) = constant, despite expansion of the Universe!(a) = constant, despite expansion of the Universe!

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It is by looking at the second equationIt is by looking at the second equationthat Supernovae people have told usthat Supernovae people have told usthe universe is accelerating!!!!!the universe is accelerating!!!!!

•• Acceleration:Acceleration:

•• Important parameter is:Important parameter is:

•• For acceleration we needFor acceleration we need

•• A Network of cosmic strings has w =A Network of cosmic strings has w = --1/3, first check1/3, first checkmore than a decade ago!more than a decade ago!

BTW here we are not assuming a lambda, itCOULD be there or a term with w also

COULD be there…

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Network of cosmic strings andNetwork of cosmic strings andtopological defects: (not examinable)topological defects: (not examinable)•• A configuration formed at a phaseA configuration formed at a phase

transition in the very earlytransition in the very earlyUniverse:Universe:

•• Can be:Can be:–– MonopolesMonopoles–– StringsStrings

D i llD i ll–– Domain wallsDomain walls–– Textures (non localised unstable)Textures (non localised unstable)

•• A mechanism for forming theseA mechanism for forming theseobjects is called the Kibbleobjects is called the Kibblemechanismmechanism

•• Could explain acceleratedCould explain acceleratedexpansion but…. Power spectrumexpansion but…. Power spectrumvery different (we will see this…)very different (we will see this…)

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Constraints from theCMB (blue) cf Supernove

Spergel et al 2003

WM

APte

am

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OUTLINE:OUTLINE:Friedmann equation derivationFriedmann equation derivationAfter the lecture, you should be able to:After the lecture, you should be able to:•• Define/derive the terms: “metric”, “scale factor”Define/derive the terms: “metric”, “scale factor”

and “coand “co--moving coordinates”moving coordinates”•• Derive the relation between the scale factor andDerive the relation between the scale factor and

redshiftredshift•• Derive the Friedman equation using NewtonianDerive the Friedman equation using Newtonian

argumentsarguments•• Describe and discuss the possible geometries ofDescribe and discuss the possible geometries of

the Universethe Universe•• [Non[Non--examinable: Discuss possible topologies ofexaminable: Discuss possible topologies of

the Universe]the Universe]

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OUTLINEOUTLINEAcceleration equation derivationAcceleration equation derivationAfter the lectures, you should be able to:After the lectures, you should be able to:•• Find the change in density as a function of scale factorFind the change in density as a function of scale factorρρ(a) for matter dominated universe, just by considering(a) for matter dominated universe, just by consideringconservation of matterconservation of matter

•• Derive the fluid equationDerive the fluid equation•• Derive the change in density as a function of scale factorDerive the change in density as a function of scale factorρρ(a) for a single fluid universe with given equation of(a) for a single fluid universe with given equation ofstate, from the fluid equationstate, from the fluid equation

•• Derive the acceleration equation from the FriedmannDerive the acceleration equation from the Friedmannequation and the fluid equationequation and the fluid equation

•• Discuss the cosmological constant and dark energyDiscuss the cosmological constant and dark energy

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END for now!!!END for now!!!